Analysis / Math Physics Seminar, Fall 2006
Friday's from 3:30 to 5 PM, meeting alternately at MIT (room 2-146) and at Harvard (Science Center room 221).
Next seminar:
Spring 2007
Full schedule:
October 3: Pierre Albin (MIT)
``Index theory on conformally compact manifolds''
October 13: Enno Lenzmann (MIT)
``Pseudo-relativistic nonlinear Schrödinger equations''
Seminar will be held at noon, in room 530 of the Harvard science center
Abstract:
This talk deals with a novel class of dispersive PDEs called
pseudo-relativisitic nonlinear Schrödinger equations.
These equations have recently found a significant application as
effective descriptions for the dynamical evolution of self-gravitating,
relativistic matter. Based on this physical motivation, I will present
results that aim at understanding the qualitative behavior of solutions
for these model equations. Here great emphasis is put on the
pseudo-relativistic Hartree equation whose focusing nonlinearity
is of critical strength in the sense that large initial data can lead to
singularity formation resulting in finite-time blow-up of the solution.
Such a breakdown indicates a ``gravitational collapse'' of the physical system
(a boson star) modeled by this equation, and it substantiates the intuitive
picture of collapsing stellar matter that exceeds a critical total mass.
If time permits, I will also discuss very recent work on
pseudo-relativistic Hartree-Fock equations which provide an effective
model for the dynamical evolution of fermion stars (such as white dwarfs).
Part of the material covered in my talk is joint work with J. Fröhlich (ETH Zürich) and L. Jonsson (KTH Stockholm).
October 20: Martin Reiris (MIT)
``A detailed discussion of the Einstein flow''
Abstract:
We will discuss in detail the existent results for the Einstein flow on the long
time. We will state theorems right from the beginning in full detail and discuss
with precision the concepts involved, sometimes commenting on the main techniques.
We will comment also on open problems (with good chance of being solved.)
October 27: Vera Hur (MIT)
``Free-surface steady water waves with vorticity''
Seminar will be held at Harvard
Abstract:
The water wave problem embodies the equation of hydrodynamics, the concepts of wave propagation, and the
critically important role of boundary dynamics. I will give a precise account of the mathematical problem describing
water waves and discuss its distinctive features. The emphasis is on the effects of the vorticity, which measures the
local swirl or eddi spin in the fluid. Existence theories of traveling waves will be presented with proofs, at least with
their ideas: Stokes waves as an application of degree theory and global bifurcation theory, and solitary waves via a
Nash-Moser implicit function theorem. I will discuss on a recent numerical simulation on th extremal rotational Stokes
waves, new types of singular configuration attributed to the vorticity. I will talk about other aspects of mathematical
study on the water waves, the well-posedness, stability, etc., if time permits.
November 3: Tai-Peng Tsai (UBC)
``Asymptotic stability of harmonic maps under the Schrödinger flow''
Seminar will be held at MIT
Abstract:
For Schrödinger maps from $\R^2\times\R^+$ to the $2$-sphere $\S^2$, it is not known if finite energy solutions can
form singularities (``blowup'') in finite time. We consider equivariant solutions with energy near the energy of the
two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four,
blowup does not occur, and global solutions converge (in a dispersive sense -- i.e. scatter) to a fixed harmonic
map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the
``generalized Hasimoto transform", and Strichartz (dispersive) estimates for a certain two space-dimensional linear
Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an
energy-space local well-posedness result for which the existence time is determined by the length-scale of a nearby
harmonic map.
This is a joint with Stephen Gustafson and Kyungkeun Kang.
November 17: CDM conference at Harvard
December 1: Gordon Ritter (Harvard)
``New Horizons in Mathematical Quantum Field Theory''
Seminar will be held at MIT
Abstract:
We will describe how the Euclidean path integral allows a
mathematically rigorous treatment of quantum field theory on an
arbitrary static spacetime manifold M. Using analytic continuation, we
will construct a unitary representation of the isometry group of M on
the Hilbert space of quantum fields propagating on the background M. The
proof is a beautiful unity between several branches of mathematics,
involving techniques from Lie theory as well as functional analysis and
geometric analysis of the Laplacian. At the end we will discuss some
open questions.
December 8: Youngmi Hur (MIT)
``Novel methodologies for effective wavelet constructions in high dimensions''
Seminar will be held at Harvard
Abstract:
I will introduce new methodologies for representing data on regular grids. Among the new methodologies,
the CAP and the L-CAMP will be discussed in detail. The CAP representation is a generalization of the
Laplacian pyramidal representation. The CAP methodology works for any spatial dimension and for any
integer dilation, and it is accompanied by solid mathematical theory that reveals its performance (i.e., its
ability to encode smoothness). The L-CAMP methodology is a descendent of the CAP. A subclass of the
L-CAMP methodology provides effective wavelet constructions in high dimensions in the sense that it has
fast algorithms (linear complexity with small constants) for both decomposition and reconstruction, its
performance is completely understood, and the sum of the volumes of the supports of the underlying mother
wavelets is extremely small. The above is joint work with Amos Ron.
I will briefly talk about work in progress on near-optimal sparsity of wavelet coefficients, if time permits.